The Autocorrelation Function and Power Spectrum of PCM/FM with Random Binary Modulating Waveforms

This paper presents the derivation of exact expressions for the autocorrelation function and power spectrum of PCM/FM or FSK when the frequency modulating waveform is a random sequence of binary pulses of length T. The problem treated is that of true frequency modulation of an oscillator, a process which, except for a few special cases, generates waveforms and spectra different from those produced by sequentially switching between the outputs of two continuously running oscillators. The final expression for the power spectrum is simple and written in closed form. The power spectrum of PCM/FM is dependent on the bit rate fB, usually defined as the reciprocal of T, and the deviation ratio D, defined as the difference between the two possible values of the instantaneous frequency divided by the bit rate. For small values of D(D<.5) the spectrum of PCM/FM has a shape resembling a high-Q resonance curve with a 3-db bandwidth given by ½π fBD2. As D increases, fB being fixed, the resonance curve becomes a poorer approximation, and with D in the vicinity of 0.7 the spectral density is nearly flat across a frequency band equal to the bit rate and drops abruptly on either side of this frequency band. It is interesting to note that a value of D near 0.7 has been found to yield minimum probability of error for given received power and receiver noise temperature. As D increases from 0.7 to 1.